Finite Group Databasebeta

← #27366148

Subgroups — #27366148

Order 13568.

936 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285108C212yesmaximal, complemented
2#20285097C211yesmaximal, characteristic, complemented
3#20274038C218yesmaximal, complemented
4#20273310C211yesmaximal, characteristic, complemented
5#20273307C212yesmaximal, complemented
6C53×#3548C211yesmaximal, characteristic
7C2×#13133217C2211yescharacteristic
8#16810993C2214yes—
9#16810982C2218yes—
10#16810975—22no—
11#16810973—22no—
12#16810969—22no—
13#16810631—22no—
14#16810624—24no—
15C4×#11719917C2211yescharacteristic
16C4×#11719917C2212yes—
17#16810365C2214yes—
18#16810363C2218yes—
19C53×#562C2214yes—
20C53×#554—22no—
21C53×#550—22no—
22C53×#547—22no—
23C53×#540—22no—
24C53×#537—24no—
25C53×#533C2211yescharacteristic
26C53×#532C2212yes—
27C22×Dic106—21no—
28C2×#11719917—22no—
29C2×#11719917—21no—
30#13133217—21no—
31#13133217—22no—
32#13133217C2312yes—
33C2×C4⋊Dic53—22no—
34#13133141—28no—
35#13133140—28no—
36#13133138—28no—
37#13133128C2312yes—
38#13133128C2314yes—
39C4×#88322—22no—
40C4×#88322—21no—
41C2×#41×C53—24no—
42C2×#41×C53D414yes—
43C4⋊Q8×C53D412yes—
44C4⋊Q8×C53D412yes—
45#127×C53—22no—
46#127×C53D412yes—
47#125×C53—22no—
48#125×C53C2312yes—
49#124×C53C2314yes—
50#123×C53—28no—
51C22⋊Q8×C53—24no—
52C22⋊Q8×C53—28no—
53C22⋊Q8×C53—22no—
54#121×C53—28no—
55#121×C53—24no—
56#121×C53—28no—
57#121×C53—22no—
58#121×C53—22no—
59C4×Q8×C53—22no—
60C4×D4×C53—24no—
61C4×D4×C53—22no—
62C4×D4×C53—22no—
63#117×C53D412yes—
64C2×C4⋊C4×C53—24no—
65C2×C42×C53C2311yescharacteristic
66#41×C53—48no—
67#41×C53—88no—
68C2×Q8×C53—22no—
69C2×Q8×C53C2×D414yes—
70C2×D4×C53—28no—
71C2×D4×C53—24no—
72C2×D4×C53—22no—
73C2×D4×C53—22no—
74C2×D4×C53C2×D414yes—
75C22×C4×C53—28no—
76C22×C4×C53—24no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D412yes—
79C22×Dic53—41no—
80C2×Dic106—22no—
81C2×Dic106—44no—
82C2×Dic106—41no—
83C4⋊C4×C53—28no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53C2×D414yes—
89C22⋊C4×C53—28no—
90C22⋊C4×C53—22no—
91C22⋊C4×C53—24no—
92C22⋊C4×C53—24no—
93C42×C53—22no—
94C42×C53C2×D411yescharacteristic
95C42×C53C2×D412yes—
96C42×C53C2411yescharacteristic
97C4⋊Dic53—24no—
98C4⋊Dic53—42no—
99#11719917—42no—
100#11719917—44no—
101C2×#88322—24no—
102C2×#88322—22no—
103C23×C53—24no—
104C23×C53#12011yescharacteristic
105Q8×C53—44no—
106Q8×C53—44no—
107D4×C53—44no—
108D4×C53—44no—
109D4×C53—44no—
110D4×C53—84no—
111D4×C53—88no—
112C2×C4×C53—22no—
113C2×C4×C53—24no—
114C2×C4×C53—24no—
115C2×C4×C53—24no—
116C2×C4×C53—48no—
117C2×C4×C53—88no—
118C2×C4×C53#12011yescharacteristic
119C2×C4×C53#12012yes—
120C2×C4×C53C22×D411yescharacteristic
121C2×C4×C53C22×D412yes—
122C2×Dic53—42no—
123C2×Dic53—82no—
124Dic106—44no—
125Dic106—82no—
126#88322—44no—
127#88322—42no—
128#19080—531nomaximal, Sylow
129C22×C53—22no—
130C22×C53—21no—
131C22×C53—44no—
132C22×C53—44no—
133C22×C53—44no—
134C22×C53C2×#12011yescharacteristic
135C4×C53—22no—
136C4×C53—21no—
137C4×C53—44no—
138C4×C53—44no—
139C4×C53#45412yes—
140C4×C53#45514yes—
141Dic53—82no—
142#3548D5311yescharacteristic
143#3141—532no—
144#3139—531no—
145#1662—538no—
146#1526—531no—
147#1525—532no—
148C2×C53—41no—
149C2×C53—84no—
150C2×C53C2×#45512yes—
151C2×C53C2×#45411yescharacteristic
152#562C2×D5314yes—
153#554—22no—
154#550—22no—
155#547—22no—
156#540—22no—
157#537—24no—
158#533C2×D5311yescharacteristic
159#532C2×D5312yes—
160C2×C4⋊Q8—531no—
161#491—534no—
162#489—538no—
163#487—1062no—
164#487—1062no—
165#486—1062no—
166#419—1062no—
167#417—1064no—
168C4×M16—531no—
169C4×M16—532no—
170#330—534no—
171#329—538no—
172C53#1908011yescharacteristic, complemented, Sylow
173C2×#41—24no—
174C2×#41C22⋊D5314yes—
175C22×Q8—1061no—
176C2×M16—1062no—
177C2×M16—1061no—
178C4⋊Q8—532no—
179C4⋊Q8—1062no—
180C4⋊Q8—1061no—
181C4⋊Q8C22⋊D5312yes—
182C4⋊Q8C22⋊D5312yes—
183#127—22no—
184#127C22⋊D5312yes—
185#125—22no—
186#125C22×D5312yes—
187#124C22×D5314yes—
188#123—28no—
189C22⋊Q8—24no—
190C22⋊Q8—28no—
191C22⋊Q8—22no—
192#121—28no—
193#121—24no—
194#121—28no—
195#121—22no—
196#121—22no—
197C4×Q8—22no—
198C4×D4—24no—
199C4×D4—22no—
200C4×D4—22no—
201#117C22⋊D5312yes—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—1062no—
204C2×C42C22×D5311yescharacteristic
205#103—1068no—
206#103—1068no—
207#102—1068no—
208#97—532no—
209#97—534no—
210C4×C8—1062no—
211C4×C8—1061no—
212#41—48no—
213#41—88no—
214C2×Q8—22no—
215C2×Q8—1062no—
216C2×Q8—2124no—
217C2×Q8—2121no—
218C2×Q8C2×C22⋊D5314yes—
219C2×D4—28no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4C2×C22⋊D5314yes—
224C22×C4—28no—
225C22×C4—24no—
226C22×C4—2121no—
227C22×C4C2×C22⋊D5312yes—
228C22×C4C2×C22⋊D5311yescharacteristic
229M16—2122no—
230M16—2124no—
231C2×C8—1064no—
232C2×C8—1062no—
233C4⋊C4—28no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—1064no—
239C4⋊C4—2122no—
240C4⋊C4C2×C22⋊D5314yes—
241C22⋊C4—28no—
242C22⋊C4—22no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C42—22no—
246C42C2×C22⋊D5311yescharacteristic
247C42C2×C22⋊D5312yes—
248C42C23×D5311yescharacteristic
249C23—24no—
250C23#1313318611yescharacteristic
251Q8—44no—
252Q8—44no—
253Q8—4242no—
254Q8—2124no—
255D4—44no—
256D4—44no—
257D4—44no—
258D4—84no—
259D4—88no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—48no—
265C2×C4—88no—
266C2×C4—4242no—
267C2×C4—2122no—
268C2×C4#1313318611yescharacteristic
269C2×C4#1313318612yes—
270C2×C4C22×C22⋊D5311yescharacteristic
271C2×C4C22×C22⋊D5312yes—
272C8—2124no—
273C8—2122no—
274C22—22no—
275C22—21no—
276C22—44no—
277C22—44no—
278C22—44no—
279C22C2×#1313318611yescharacteristic
280C4—22no—
281C4—21no—
282C4—44no—
283C4—44no—
284C4—4242no—
285C4#1681079612yes—
286C4#1681080114yes—
287C2—41no—
288C2—84no—
289C2C2×#1681080112yes—
290C2C2×#1681079611yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3548C2
socleC22×C53C2×#120
hypercenter#3548D53
hypocenterC53#19080
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20285108C2semidirect
2. #20285097C2semidirect
3. #20274038C2semidirect
4. #20273310C2semidirect
5. #20273307C2semidirect
172. C53#19080semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202851082
#202850972
#202740382
#202733102
#202733072
C53×#35482
#1908053